Moduli Spaces of Embedded Constant Mean Curvature Surfaces with Few Ends and Special Symmetry

TL;DR

This paper investigates the structure and classification of complete embedded constant mean curvature surfaces with three or four ends, focusing on symmetry constraints and their moduli spaces, using geometric and topological methods.

Contribution

It provides necessary conditions and characterizations for specific CMC surfaces with few ends and symmetries, enriching the understanding of their moduli spaces.

Findings

Respective submoduli spaces are two-dimensional varieties.

Fundamental domains characterized by great circle polygons in the three-sphere.

Necessary conditions for surfaces with three or four ends under reflection symmetries.

Abstract

We give necessary conditions on complete embedded \cmc surfaces with three or four ends subject to reflection symmetries. The respective submoduli spaces are two-dimensional varieties in the moduli spaces of general \cmc surfaces. We characterize fundamental domains of our \cmc surfaces by associated great circle polygons in the three-sphere.